Lectures on Statistical Data Analysis

1. Lectures on Statistical Data Analysis RWTH Aachen Graduiertenkolleg 12-16 February, 2007 Glen Cowan Physics Department Royal Holloway, University of London g.cowan@rhul.ac.uk www.pp.rhul.ac.uk/~cowan Course web page: www.pp.rhul.ac.uk/~cowan/stat_aachen.html Lectures on Statistical Data Analysis

2. Outline by day (approx.) 1 Probability Definition, Bayes’ theorem, probability densities and their properties, catalogue of pdfs, Monte Carlo 2 Statistical tests general concepts, test statistics, multivariate methods, goodness-of-fit tests 3 Parameter estimation general concepts, maximum likelihood, variance of estimators, least squares 4 Interval estimation setting limits 5 Further topics systematic errors, MCMC, unfolding Lectures on Statistical Data Analysis

3. Some statistics books, papers, etc. G. Cowan, Statistical Data Analysis, Clarendon, Oxford, 1998 see also www.pp.rhul.ac.uk/~cowan/sda R.J. Barlow, Statistics, A Guide to the Use of Statistical in the Physical Sciences, Wiley, 1989 see also hepwww.ph.man.ac.uk/~roger/book.html L. Lyons, Statistics for Nuclear and Particle Physics, CUP, 1986 W. Eadie et al., Statistical and Computational Methods in Experimental Physics, North-Holland, 1971 (2nd ed. imminent) S. Brandt, Statistical and Computational Methods in Data Analysis, Springer, New York, 1998 (with program library on CD) W.M. Yao et al. (Particle Data Group), Review of Particle Physics, Journal of Physics G 33 (2006) 1; see also pdg.lbl.gov sections on probability statistics, Monte Carlo Lectures on Statistical Data Analysis

4. Data analysis in particle physics Observe events of a certain type Measure characteristics of each event (particle momenta, number of muons, energy of jets,...) Theories (e.g. SM) predict distributions of these properties up to free parameters, e.g., a, GF, MZ, as, mH, ... Some tasks of data analysis: Estimate (measure) the parameters; Quantify the uncertainty of the parameter estimates; Test the extent to which the predictions of a theory are in agreement with the data. Lectures on Statistical Data Analysis

5. Dealing with uncertainty In particle physics there are various elements of uncertainty: theory is not deterministic quantum mechanics random measurement errors present even without quantum effects things we could know in principle but don’t e.g. from limitations of cost, time, ... We can quantify the uncertainty usingPROBABILITY Lectures on Statistical Data Analysis

6. A definition of probability Consider a set S with subsets A, B, ... Kolmogorov axioms (1933) From these axioms we can derive further properties, e.g. Lectures on Statistical Data Analysis

7. Conditional probability, independence Also define conditional probability of A given B (with P(B) ≠ 0): E.g. rolling dice: Subsets A, Bindependent if: If A, B independent, N.B. do not confuse with disjoint subsets, i.e., Lectures on Statistical Data Analysis

8. Interpretation of probability I. Relative frequency A, B, ... are outcomes of a repeatable experiment cf. quantum mechanics, particle scattering, radioactive decay... II. Subjective probability A, B, ... are hypotheses (statements that are true or false) • Both interpretations consistent with Kolmogorov axioms. • In particle physics frequency interpretation often most useful, but subjective probability can provide more natural treatment of non-repeatable phenomena: systematic uncertainties, probability that Higgs boson exists,... Lectures on Statistical Data Analysis

9. Bayes’ theorem From the definition of conditional probability we have and , so but Bayes’ theorem First published (posthumously) by the Reverend Thomas Bayes (1702−1761) An essay towards solving a problem in the doctrine of chances, Philos. Trans. R. Soc. 53 (1763) 370; reprinted in Biometrika, 45 (1958) 293. Lectures on Statistical Data Analysis

10. B The law of total probability Consider a subset B of the sample space S, S divided into disjoint subsets Ai such that [i Ai = S, Ai B∩ Ai → → law of total probability → Bayes’ theorem becomes Lectures on Statistical Data Analysis

11. An example using Bayes’ theorem Suppose the probability (for anyone) to have AIDS is: ←prior probabilities, i.e., before any test carried out Consider an AIDS test: result is + or - ←probabilities to (in)correctly identify an infected person ←probabilities to (in)correctly identify an uninfected person Suppose your result is +. How worried should you be? Lectures on Statistical Data Analysis

12. Bayes’ theorem example (cont.) The probability to have AIDS given a + result is ← posterior probability i.e. you’re probably OK! Your viewpoint: my degree of belief that I have AIDS is 3.2% Your doctor’s viewpoint: 3.2% of people like this will have AIDS Lectures on Statistical Data Analysis

13. Frequentist Statistics − general philosophy In frequentist statistics, probabilities are associated only with the data, i.e., outcomes of repeatable observations (shorthand: ). Probability = limiting frequency Probabilities such as P (Higgs boson exists), P (0.117 < as < 0.121), etc. are either 0 or 1, but we don’t know which. The tools of frequentist statistics tell us what to expect, under the assumption of certain probabilities, about hypothetical repeated observations. The preferred theories (models, hypotheses, ...) are those for which our observations would be considered ‘usual’. Lectures on Statistical Data Analysis

14. Bayesian Statistics − general philosophy In Bayesian statistics, use subjective probability for hypotheses: probability of the data assuming hypothesis H (the likelihood) prior probability, i.e., before seeing the data posterior probability, i.e., after seeing the data normalization involves sum over all possible hypotheses Bayes’ theorem has an “if-then” character: If your prior probabilities were p (H), then it says how these probabilities should change in the light of the data. No general prescription for priors (subjective!) Lectures on Statistical Data Analysis

15. Random variables and probability density functions A random variable is a numerical characteristic assigned to an element of the sample space; can be discrete or continuous. Suppose outcome of experiment is continuous value x →f(x) = probability density function (pdf) x must be somewhere Or for discrete outcome xi with e.g. i = 1, 2, ... we have probability mass function x must take on one of its possible values Lectures on Statistical Data Analysis

16. Cumulative distribution function Probability to have outcome less than or equal to x is cumulative distribution function Alternatively define pdf with Lectures on Statistical Data Analysis

17. Other types of probability densities Outcome of experiment characterized by several values, e.g. an n-component vector, (x1, ... xn) →joint pdf Sometimes we want only pdf of some (or one) of the components →marginal pdf x1, x2 independent if Sometimes we want to consider some components as constant →conditional pdf Lectures on Statistical Data Analysis

18. Expectation values Consider continuous r.v. x with pdf f (x). Define expectation (mean) value as Notation (often): ~ “centre of gravity” of pdf. For a function y(x) with pdf g(y), (equivalent) Variance: Notation: Standard deviation: s ~ width of pdf, same units as x. Lectures on Statistical Data Analysis

19. Covariance and correlation Define covariance cov[x,y] (also use matrix notation Vxy) as Correlation coefficient (dimensionless) defined as If x, y, independent, i.e., , then → x and y, ‘uncorrelated’ N.B. converse not always true. Lectures on Statistical Data Analysis

20. Correlation (cont.) Lectures on Statistical Data Analysis

21. Error propagation Suppose we measure a set of values and we have the covariances which quantify the measurement errors in the xi. Now consider a function What is the variance of to find the pdf The hard way: use joint pdf then from g(y) find V[y] = E[y2] - (E[y])2. may not even be fully known. Often not practical, Lectures on Statistical Data Analysis

22. Error propagation (2) Suppose we had in practice only estimates given by the measured Expand to 1st order in a Taylor series about To find V[y] we need E[y2] and E[y]. since Lectures on Statistical Data Analysis

23. Error propagation (3) Putting the ingredients together gives the variance of Lectures on Statistical Data Analysis

24. Error propagation (4) If the xi are uncorrelated, i.e., then this becomes Similar for a set of m functions or in matrix notation where Lectures on Statistical Data Analysis

25. Error propagation (5) y(x) The ‘error propagation’ formulae tell us the covariances of a set of functions in terms of the covariances of the original variables. sy x sx Limitations: exact only if linear. y(x) Approximation breaks down if function nonlinear over a region comparable in size to the si. ? x sx N.B. We have said nothing about the exact pdf of the xi, e.g., it doesn’t have to be Gaussian. Lectures on Statistical Data Analysis

26. Error propagation − special cases → → That is, if the xi are uncorrelated: add errors quadratically for the sum (or difference), add relative errors quadratically for product (or ratio). But correlations can change this completely... Lectures on Statistical Data Analysis

27. Error propagation − special cases (2) Consider with Now suppose r = 1. Then i.e. for 100% correlation, error in difference → 0. Lectures on Statistical Data Analysis

28. Some distributions Distribution/pdf Example use in HEP Binomial Branching ratio Multinomial Histogram with fixed N Poisson Number of events found Uniform Monte Carlo method Exponential Decay time Gaussian Measurement error Chi-square Goodness-of-fit Cauchy Mass of resonance Landau Ionization energy loss Lectures on Statistical Data Analysis

29. Binomial distribution Consider N independent experiments (Bernoulli trials): outcome of each is ‘success’ or ‘failure’, probability of success on any given trial is p. Define discrete r.v. n = number of successes (0 ≤n ≤ N). Probability of a specific outcome (in order), e.g. ‘ssfsf’ is But order not important; there are ways (permutations) to get n successes in N trials, total probability for n is sum of probabilities for each permutation. Lectures on Statistical Data Analysis

30. Binomial distribution (2) The binomial distribution is therefore parameters random variable For the expectation value and variance we find: Lectures on Statistical Data Analysis

31. Binomial distribution (3) Binomial distribution for several values of the parameters: Example: observe N decays of W±, the number n of which are W→mn is a binomial r.v., p = branching ratio. Lectures on Statistical Data Analysis

32. Multinomial distribution Like binomial but now m outcomes instead of two, probabilities are For N trials we want the probability to obtain: n1 of outcome 1, n2 of outcome 2,  nm of outcome m. This is the multinomial distribution for Lectures on Statistical Data Analysis

33. Multinomial distribution (2) Now consider outcome i as ‘success’, all others as ‘failure’. → all ni individually binomial with parameters N, pi for all i One can also find the covariance to be represents a histogram Example: with m bins, N total entries, all entries independent. Lectures on Statistical Data Analysis Glen Cowan CERN Summer Student Lectures on Statistics

34. Poisson distribution Consider binomial n in the limit →n follows the Poisson distribution: Example: number of scattering events n with cross section s found for a fixed integrated luminosity, with Lectures on Statistical Data Analysis

35. Uniform distribution Consider a continuous r.v. x with -∞ < x < ∞ . Uniform pdf is: N.B. For any r.v. x with cumulative distribution F(x), y = F(x) is uniform in [0,1]. Example: for p0→gg, Eg is uniform in [Emin, Emax], with Lectures on Statistical Data Analysis

36. Exponential distribution The exponential pdf for the continuous r.v. x is defined by: Example: proper decay time t of an unstable particle (t = mean lifetime) Lack of memory (unique to exponential): Lectures on Statistical Data Analysis

37. Gaussian distribution The Gaussian (normal) pdf for a continuous r.v. x is defined by: (N.B. often m, s2 denote mean, variance of any r.v., not only Gaussian.) Special case: m = 0, s2 = 1 (‘standard Gaussian’): If y ~ Gaussian with m, s2, then x = (y-m) /s follows  (x). Lectures on Statistical Data Analysis

38. Gaussian pdf and the Central Limit Theorem The Gaussian pdf is so useful because almost any random variable that is a sum of a large number of small contributions follows it. This follows from the Central Limit Theorem: For n independent r.v.s xi with finite variances si2, otherwise arbitrary pdfs, consider the sum In the limit n→ ∞, y is a Gaussian r.v. with Measurement errors are often the sum of many contributions, so frequently measured values can be treated as Gaussian r.v.s. Lectures on Statistical Data Analysis

39. Central Limit Theorem (2) The CLT can be proved using characteristic functions (Fourier transforms), see, e.g., SDA Chapter 10. For finite n, the theorem is approximately valid to the extent that the fluctuation of the sum is not dominated by one (or few) terms. Beware of measurement errors with non-Gaussian tails. Good example: velocity component vx of air molecules. OK example: total deflection due to multiple Coulomb scattering. (Rare large angle deflections give non-Gaussian tail.) Bad example: energy loss of charged particle traversing thin gas layer. (Rare collisions make up large fraction of energy loss, cf. Landau pdf.) Lectures on Statistical Data Analysis

40. Multivariate Gaussian distribution Multivariate Gaussian pdf for the vector are transpose (row) vectors, are column vectors, For n = 2 this is where r = cov[x1, x2]/(s1s2) is the correlation coefficient. Lectures on Statistical Data Analysis

41. Chi-square (c2) distribution The chi-square pdf for the continuous r.v. z (z≥ 0) is defined by n = 1, 2, ... = number of ‘degrees of freedom’ (dof) For independent Gaussian xi, i = 1, ..., n, means mi, variances si2, follows c2 pdf with n dof. Example: goodness-of-fit test variable especially in conjunction with method of least squares. Lectures on Statistical Data Analysis

42. Cauchy (Breit-Wigner) distribution The Breit-Wigner pdf for the continuous r.v. xis defined by (G = 2, x0 = 0 is the Cauchy pdf.) E[x] not well defined, V[x] →∞. x0 = mode (most probable value) G = full width at half maximum Example: mass of resonance particle, e.g. r, K*, f0, ... G = decay rate (inverse of mean lifetime) Lectures on Statistical Data Analysis

43. Landau distribution For a charged particle with b = v /c traversing a layer of matter of thickness d, the energy loss D follows the Landau pdf: D + - + - b - + - + d L. Landau, J. Phys. USSR 8 (1944) 201; see also W. Allison and J. Cobb, Ann. Rev. Nucl. Part. Sci. 30 (1980) 253. Lectures on Statistical Data Analysis

44. Landau distribution (2) Long ‘Landau tail’ →all moments ∞ Mode (most probable value) sensitive to b , →particle i.d. Lectures on Statistical Data Analysis

45. The Monte Carlo method What it is: a numerical technique for calculating probabilities and related quantities using sequences of random numbers. The usual steps: (1) Generate sequence r1, r2, ..., rm uniform in [0, 1]. (2) Use this to produce another sequence x1, x2, ..., xn distributed according to some pdf f (x) in which we’re interested (x can be a vector). (3) Use the x values to estimate some property of f (x), e.g., fraction of x values with a < x < b gives → MC calculation = integration (at least formally) MC generated values = ‘simulated data’ →use for testing statistical procedures Lectures on Statistical Data Analysis

46. Random number generators Goal: generate uniformly distributed values in [0, 1]. Toss coin for e.g. 32 bit number... (too tiring). → ‘random number generator’ = computer algorithm to generate r1, r2, ..., rn. Example: multiplicative linear congruential generator (MLCG) ni+1 = (a ni) mod m , where ni = integer a = multiplier m = modulus n0 = seed (initial value) N.B. mod = modulus (remainder), e.g. 27 mod 5 = 2. This rule produces a sequence of numbers n0, n1, ... Lectures on Statistical Data Analysis

47. Random number generators (2) The sequence is (unfortunately) periodic! Example (see Brandt Ch 4): a = 3, m = 7, n0 = 1 ← sequence repeats Choose a, m to obtain long period (maximum = m- 1); m usually close to the largest integer that can represented in the computer. Only use a subset of a single period of the sequence. Lectures on Statistical Data Analysis

48. Random number generators (3) are in [0, 1] but are they ‘random’? Choose a, m so that the ri pass various tests of randomness: uniform distribution in [0, 1], all values independent (no correlations between pairs), e.g. L’Ecuyer, Commun. ACM 31 (1988) 742 suggests a = 40692 m = 2147483399 Far better algorithms available, e.g. RANMAR, period See F. James, Comp. Phys. Comm. 60 (1990) 111; Brandt Ch. 4 Lectures on Statistical Data Analysis

49. The transformation method Given r1, r2,..., rn uniform in [0, 1], find x1, x2,..., xn that follow f (x) by finding a suitable transformation x (r). Require: i.e. That is, set and solve for x (r). Lectures on Statistical Data Analysis

50. Example of the transformation method Exponential pdf: Set and solve for x (r). works too.) → Lectures on Statistical Data Analysis